# Safety Certification for Stochastic Systems via Neural Barrier Functions

arXiv: https://arxiv.org/abs/2206.01463 · HTML: https://mlsanity.com/paper/2206.01463/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 49 references that are in the map.
- First posted on arXiv 3 Jun 2022.

## Builds on (most cited first, 21 of 21)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Probabilistic Safety for Bayesian Neural Networks](https://mlsanity.com/paper/2004.10281/) (2020)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Bayesian Inference with Certifiable Adversarial Robustness](https://mlsanity.com/paper/2102.05289/) (2021)
- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Formal Synthesis of Stochastic Systems via Control Barrier Certificates](https://mlsanity.com/paper/1905.04585/) (2019)
- [A Barrier Function Approach to Finite-Time Stochastic System Verification and Control](https://mlsanity.com/paper/1909.05109/) (2019)
- [Efficiency through Uncertainty: Scalable Formal Synthesis for Stochastic Hybrid Systems](https://mlsanity.com/paper/1901.01576/) (2019)
- [Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systems](https://mlsanity.com/paper/2203.11812/) (2022)
- [Strategy Synthesis for Partially-known Switched Stochastic Systems](https://mlsanity.com/paper/2104.02172/) (2021)

## Built on by (most cited first, 22 of 22)

- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [Steering Dialogue Dynamics for Robustness against Multi-turn Jailbreaking Attacks](https://mlsanity.com/paper/2503.00187/) (2025)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [Compositional Policy Learning in Stochastic Control Systems with Formal Guarantees](https://mlsanity.com/paper/2312.01456/) (2023)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Unifying Qualitative and Quantitative Safety Verification of DNN-Controlled Systems](https://mlsanity.com/paper/2404.01769/) (2024)
- [Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates](https://mlsanity.com/paper/2406.00826/) (2024)
- [Data-Driven Permissible Safe Control with Barrier Certificates](https://mlsanity.com/paper/2405.00136/) (2024)
- [Scalable Verification of Neural Control Barrier Functions Using Linear Bound Propagation](https://mlsanity.com/paper/2511.06341/) (2025)
- [Online Efficient Safety-Critical Control for Mobile Robots in Unknown Dynamic Multi-Obstacle Environments](https://mlsanity.com/paper/2402.16449/) (2024)
- [Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model](https://mlsanity.com/paper/2405.13735/) (2024)
- [Dynamic Tube MPC: Learning Tube Dynamics with Massively Parallel Simulation for Robust Safety in Practice](https://mlsanity.com/paper/2411.15350/) (2024)
- [Efficient Exploration Using Extra Safety Budget in Constrained Policy Optimization](https://mlsanity.com/paper/2302.14339/) (2023)
- [Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks](https://mlsanity.com/paper/2402.18677/) (2024)
- [VeRecycle: Reclaiming Guarantees from Probabilistic Certificates for Stochastic Dynamical Systems after Change](https://mlsanity.com/paper/2505.14001/) (2025)
- [Set Invariance with Probability One for Controlled Diffusion: Score-based Approach](https://mlsanity.com/paper/2507.22385/) (2025)
- [Formal Synthesis of Certifiably Robust Neural Lyapunov-Barrier Certificates](https://mlsanity.com/paper/2602.05311/) (2026)
- [Efficient Verification of Neural Control Barrier Functions with Smooth Nonlinear Activations](https://mlsanity.com/paper/2605.07757/) (2026)
- [k-Inductive Neural Barrier Certificates for Unknown Nonlinear Dynamics](https://mlsanity.com/paper/2605.20108/) (2026)
- [Certificate-Guided Evaluation of Reinforcement Learning Generalization](https://mlsanity.com/paper/2606.00840/) (2026)

## Abstract (arXiv)

Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem that limits the wider adoption of autonomous systems in safety-critical applications. One promising solution to address this problem is barrier functions. The composition of a barrier function with a stochastic system forms a supermartingale, thus enabling the computation of the probability that the system stays in a safe set over a finite time horizon via martingale inequalities. However, existing approaches to find barrier functions for stochastic systems generally rely on convex optimization programs that restrict the search of a barrier to a small class of functions such as low degree SoS polynomials and can be computationally expensive. In this paper, we parameterize a barrier function as a neural network and show that techniques for robust training of neural networks can be successfully employed to find neural barrier functions. Specifically, we leverage bound propagation techniques to certify that a neural network satisfies the conditions to be a barrier function via linear programming and then employ the resulting bounds at training time to enforce the satisfaction of these conditions. We also present a branch-and-bound scheme that makes the certification framework scalable. We show that our approach outperforms existing methods in several case studies and often returns certificates of safety that are orders of magnitude larger.